So much fun to read this, Jack! Who else would've thought of this? But then again, being unable to help my disgracefully pedantic self, and at serious risk of becoming "that guy," I have to wonder: when the AI did its calculations, what were its assumptions about the masses of the two objects? Did it assume that the Sun and Earth, upon being shrunk to the size of your hand, would retain their entire full-size masses, the way a neutron star does when it collapses? Or, knowing that mass grows as the cube of the increase in linear dimensions (e.g. a two-inch cube has 8 times the mass of a one-inch cube of the same material) and shrinks as the cube root of linear dimensions, did the AI assume a vastly reduced mass for the two spheroids on the equally reasonable grounds that we're just talking about two much smaller objects, not things compressed by gravitational forces? Something tells me that might have a huge effect on the outcome.... If I had to guess, I'd guess it's the former assumption, and the resulting implications are truly mind-boggling (an atomic bomb in your face being a billion times dimmer than a supernova located where the Sun is now. Damn, what a thought!)
Hi Rip – the calculations were based on the masses remaining the same and only the physical dimensions of the system changing. The mass of a black hole, for instance, of one solar mass, is exactly one solar mass, albeit it is more compact
Gotcha, thanks! I wonder, if assumption number two applied, would it even be possible for the low-mass mini-Earth to orbit the low-mass mini-Sun? And if so, what would the orbital radius be, and what would the orbital period be? Or is the gravitational force, at those masses, so small that the very concept of an orbiting body becomes impossible?
Vell sir, the operating assumption for the story is that the masses of the objects remain the same, and gravity is a pretty weak force so if you have an object in let us say free fall that is 0.5mm in diameter and two inches away there's an object roughly the size of a human blood cell, I don't think they would be coupled nearly strongly enough for a Newtonian/Keplerian orbit – the gravitational attraction wouldn't be zero but I think not nearly strong enough. I mean keep in mind that gravity is so weak that it takes the entire Earth to produce enough gravitational force to hold you onto the surface. I cheated a little and asked ChatGPT to calculate the gravity produced by an 0.5mm grain of sand and at a distance of one mm, the force is about one six-trillionth of a newton, which is well below the detection threshold of any known instrument. In fact the gravitational field is so weak that the escape velocity from the sand grain can be incredibly slow – on the order of a couple of nanometers per second.
Danke, Herr Professor Doktor! I guess that leads to yet another question (aren't you glad you opened this particular Pandora's Box? Or maybe that I did...): in practical terms, what would be the approximate minimum masses required for one object to have sufficient gravitational "oomph" to keep another object in orbit around it? Kinda like those Russian nesting dolls: we know that our own moon could easily sustain a smaller object in orbit around itself, even though apparently no such moon's moon exists--we know because the various Apollo command modules entered orbit around the moon, so presumably a much more massive celestial body could do so also. And could THAT smaller object, in turn, sustain its own smaller satellite? (Let's ignore the obvious three-body problems that this might lead to and assume each pair exists in a vacuum, so to speak). Etc, etc, etc--where does it end? Where, practically speaking, do two masses become so small that orbit around the larger one cannot be achieved or sustained? (PS feel free to treat this as rhetorical pondering, and forgive my lack of AI fluency which prevents me from just finding the answer to the damn question myself....)
Well just to keep from leaning on ChatGPT too much, I would say small but not, realistically, arbitrarily small – the smallest object in the Solar System known to have a "moon" is the asteroid Didymos, which is 780 meters in diameter and which has the smaller asteroid, Dimorphos, in orbit around it. For an object to have enough gravity for another object to have a stable orbit around it, there has to be enough gravity to create a sufficient centripedal force to maintain the orbit and below a certain amount of gravity, the necessary orbital speed is so slow that practically speaking an orbit is impossible. For our hypothetical sand grain, for instance, if the escape velocity is just a couple of nanometers per second then the necessary orbital speed would be below that – for scale, a hydrogen atom is about a tenth of a nanometer in diameter. At that scale I suspect quantum fluctuations would be enough to prevent a stable orbit from forming. How small is too small? I suppose that depends on the masses of the primary and orbiting bodies; I wonder if at some point a binary system isn't more stable. In a situation with point masses and absolutely no other factors I suppose you could actually have orbits with masses that are arbitrarily small, although that is an idealized situation – a pendulum swinging in a perfect vacuum on a frictionless bearing will swing forever, or almost, but then in this idealized situation you have to ignore the fact that the pendulum is losing energy slowly through gravity waves. Minute effect but not nothing and forever is a long time.
No doubt an astrophysics PhD candidate has written a dissertation on something like this question. "Baseballs Orbiting Great Pyramids: An Exploration of the Smallest Attainable Orbital Pair Under Real-World Conditions."
You can take the watch man out of the physics class, but not the physics class out of the watch man.
đŸ˜‚ too true
So much fun to read this, Jack! Who else would've thought of this? But then again, being unable to help my disgracefully pedantic self, and at serious risk of becoming "that guy," I have to wonder: when the AI did its calculations, what were its assumptions about the masses of the two objects? Did it assume that the Sun and Earth, upon being shrunk to the size of your hand, would retain their entire full-size masses, the way a neutron star does when it collapses? Or, knowing that mass grows as the cube of the increase in linear dimensions (e.g. a two-inch cube has 8 times the mass of a one-inch cube of the same material) and shrinks as the cube root of linear dimensions, did the AI assume a vastly reduced mass for the two spheroids on the equally reasonable grounds that we're just talking about two much smaller objects, not things compressed by gravitational forces? Something tells me that might have a huge effect on the outcome.... If I had to guess, I'd guess it's the former assumption, and the resulting implications are truly mind-boggling (an atomic bomb in your face being a billion times dimmer than a supernova located where the Sun is now. Damn, what a thought!)
Hi Rip – the calculations were based on the masses remaining the same and only the physical dimensions of the system changing. The mass of a black hole, for instance, of one solar mass, is exactly one solar mass, albeit it is more compact
Gotcha, thanks! I wonder, if assumption number two applied, would it even be possible for the low-mass mini-Earth to orbit the low-mass mini-Sun? And if so, what would the orbital radius be, and what would the orbital period be? Or is the gravitational force, at those masses, so small that the very concept of an orbiting body becomes impossible?
Vell sir, the operating assumption for the story is that the masses of the objects remain the same, and gravity is a pretty weak force so if you have an object in let us say free fall that is 0.5mm in diameter and two inches away there's an object roughly the size of a human blood cell, I don't think they would be coupled nearly strongly enough for a Newtonian/Keplerian orbit – the gravitational attraction wouldn't be zero but I think not nearly strong enough. I mean keep in mind that gravity is so weak that it takes the entire Earth to produce enough gravitational force to hold you onto the surface. I cheated a little and asked ChatGPT to calculate the gravity produced by an 0.5mm grain of sand and at a distance of one mm, the force is about one six-trillionth of a newton, which is well below the detection threshold of any known instrument. In fact the gravitational field is so weak that the escape velocity from the sand grain can be incredibly slow – on the order of a couple of nanometers per second.
Danke, Herr Professor Doktor! I guess that leads to yet another question (aren't you glad you opened this particular Pandora's Box? Or maybe that I did...): in practical terms, what would be the approximate minimum masses required for one object to have sufficient gravitational "oomph" to keep another object in orbit around it? Kinda like those Russian nesting dolls: we know that our own moon could easily sustain a smaller object in orbit around itself, even though apparently no such moon's moon exists--we know because the various Apollo command modules entered orbit around the moon, so presumably a much more massive celestial body could do so also. And could THAT smaller object, in turn, sustain its own smaller satellite? (Let's ignore the obvious three-body problems that this might lead to and assume each pair exists in a vacuum, so to speak). Etc, etc, etc--where does it end? Where, practically speaking, do two masses become so small that orbit around the larger one cannot be achieved or sustained? (PS feel free to treat this as rhetorical pondering, and forgive my lack of AI fluency which prevents me from just finding the answer to the damn question myself....)
Well just to keep from leaning on ChatGPT too much, I would say small but not, realistically, arbitrarily small – the smallest object in the Solar System known to have a "moon" is the asteroid Didymos, which is 780 meters in diameter and which has the smaller asteroid, Dimorphos, in orbit around it. For an object to have enough gravity for another object to have a stable orbit around it, there has to be enough gravity to create a sufficient centripedal force to maintain the orbit and below a certain amount of gravity, the necessary orbital speed is so slow that practically speaking an orbit is impossible. For our hypothetical sand grain, for instance, if the escape velocity is just a couple of nanometers per second then the necessary orbital speed would be below that – for scale, a hydrogen atom is about a tenth of a nanometer in diameter. At that scale I suspect quantum fluctuations would be enough to prevent a stable orbit from forming. How small is too small? I suppose that depends on the masses of the primary and orbiting bodies; I wonder if at some point a binary system isn't more stable. In a situation with point masses and absolutely no other factors I suppose you could actually have orbits with masses that are arbitrarily small, although that is an idealized situation – a pendulum swinging in a perfect vacuum on a frictionless bearing will swing forever, or almost, but then in this idealized situation you have to ignore the fact that the pendulum is losing energy slowly through gravity waves. Minute effect but not nothing and forever is a long time.
No doubt an astrophysics PhD candidate has written a dissertation on something like this question. "Baseballs Orbiting Great Pyramids: An Exploration of the Smallest Attainable Orbital Pair Under Real-World Conditions."